Motion in a Circle With Non Constant Acceleration

Suppose a particle is moving on a circle of radius  
\[r\]
  with at any time  
\[t\]
  the ant clockwise angle of the particle from the positive  
\[x\]
  axis being given by  
\[\theta = \frac{t}{t+1}\]
/>br /> the angular velocity is  
\[\omega = \frac{d \theta }{dt} =\frac{(t+1) \frac{d(t)}{dt}- t \frac{d(t+1)}{dt}}{(t+1)^2}= \frac{1}{(t+1)^2}\]

using the quotient rule.
Similarly the angular acceleration is  
\[\alpha = - \frac{2}{(t+1)^2}\]
.
The distance moved is a time  
\[t\]
  is  
\[s= r \theta = =\frac{rt}{t+1}\]
, the speed is  
\[v= r \omega = \frac{r}{(t+1)^2}\]
  and the acceleration is  
\[a= r \alpha = - \frac{2r}{(r+1)^3}\]
.
As  
\[t \rightarrow \infty, \; \theta \rightarrow 1\]
  and  
\[\theta , \alpha \rightarrow 0\]
.

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